Birthdays: Probability, Statistics and Data Visualisation

Zak Varty

Question

Talk to your neighbour:


  • What is the probability that two or more people in this room share a birthday?


  • What assumptions are you making to arrive at your answer?
03:00

Typical Assumptions


  • No twins, triplets, etc. (birth dates are independent)
  • All days are equally probable.
  • There are 365 days in the year.
  • Pr(Shared birthday) = 1 - Pr(\(n\) distinct birthdays).


\[\begin{align} \text{Pr}(n \text{ distinct birthdays}) &= \frac{365}{365} \times \frac{364}{365} \times \dots \times \frac{365 - (n -1)}{365} \\ & \\ &= \prod_{i=0}^{n-1} \left\{\frac{365-i}{365}\right\}. \end{align}\]

Plotting this Solution

Testing Our Assumptions

Are all days really equally likely?

Correcting for Feb 29

A different view of the same data

Handling holidays

  • Only emergency treatments on Christmas, Boxing day and New Years Day.
  • Treat these days separately. (Skip over hypothesis tests here)


\[\widehat p = \frac{\text{relative count}}{\text{total relative count}}\]

\[\widehat{p}_{01/01} = 0.00237 \approx \frac{1}{421}, \quad \widehat{p}_{25/12} = 0.00214 \approx \frac{1}{467}, \quad \widehat{p}_{26/12} = 0.00204 \approx \frac{1}{490}.\]

What about the non-holidays?

But we have all of this information!

What are we testing?



\[ H_0: \text{all non-holidays have the same mean birth count.}\]

\[vs.\]

\[H_1: \text{At least one day has a different mean.}\]

Bootstrap confidence interval - idea


  • Under \(H_0\) and the observations being i.i.d., the observed counts could have happened on any day
  • We can build another day that we have not observed based on our data
  • Build lots of these days and calculate the mean of each one
  • Use the quantiles of these e.g. \((\bar x_{0.025}, \bar x_{0.975})\) to construct a confidence interval.

Bootstrap confidence interval - procedure

  • Sample 22 non holiday birth counts at random, with replacement.
  • Calculate their mean.
  • Add this to a vector of means.
  • Repeat until we have 10,000 means.
  • Calculate the desired quantiles of the mean vector.
  • Compare to observations.

Bootstrap confidence interval - code

# 1: set up data & storage
boot_data <- as.vector(as.matrix(non_hols[,3:24]))
boot_sample <- rep(NA,22)

boot_size <- 1e5
boot_means  <- rep(NA,boot_size) 

# 2: make fake data sets where day does not matter
for (i in 1:boot_size) {
  boot_sample <- sample(boot_data, size = 22, replace = TRUE)
  boot_means[i] <- mean(boot_sample,na.rm = TRUE)
}

# 3: find "typical" range if day does not influence count
boot_CI <- quantile(boot_means, probs = c(0.025,0.975))
boot_CI
    2.5%    97.5% 
1737.091 1909.900

Bootstrap confidence interval - results

Other questions

We could take this analysis much further:


  • Is there dependence on the previous count(s)?
  • Is there an effect from the day of week?
  • Is there an trend over time?


If you want to learn more: https://pudding.cool/2018/04/birthday-paradox/

Takeaways



  • Statistics is about understanding the assumptions.
  • The more assumptions you make the more you can conclude.
  • Display choices impact what is seen and what is hidden.
  • A statistician’s work is never done!

Any quick questions? I’ll be around at coffee for longer ones!

::: ::::

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